Source Redshifts from Gravitational-Wave Observations of Binary Neutron Star Mergers

Inspiraling compact binaries as standard sirens will become an invaluable tool for cosmology when we enter the gravitational-wave detection era. However, a degeneracy in the information carried by gravitational waves between the total rest-frame mass $M$ and the redshift $z$ of the source implies that neither can be directly extracted from the signal; only the combination $M(1+z)$, the redshifted mass, can be directly extracted from the signal. Recent work has shown that for third-generation detectors, a tidal correction to the gravitational-wave phase in the late-inspiral signal of binary neutron star systems can be used to break the mass-redshift degeneracy. Here, we propose to use the signature encoded in the postmerger signal allowing the accurate extraction of the intrinsic rest-frame mass of the source, in turn permitting the determination of source redshift and luminosity distance. The entirety of this analysis method and any subsequent cosmological inference derived from it would be obtained solely from gravitational-wave observations and, hence, would be independent of the cosmological distance ladder. Using numerical simulations of binary neutron star mergers of different mass, we model gravitational-wave signals at different redshifts and use a Bayesian parameter estimation to determine the accuracy with which the redshift and mass can be extracted. We find that for a known illustrative neutron star equation of state and using the Einstein telescope, the median of the $1\sigma$ confidence regions in redshift corresponds to $\sim 10\%–20\%$ uncertainties at redshifts of $z<0.04$.

Popular Summary

According to Einstein’s theory of general relativity, the acceleration of mass leads to the emission of energy in the form of gravitational radiation. There is currently a global effort underway to detect this radiation from astrophysical sources. Some of the most likely sources of gravitational radiation are inspiraling binary neutron stars; upcoming detectors should be able to detect approximately ten merger events per year. To accurately model the dynamics of such systems and to compute the emitted gravitational radiation, we rely on cutting-edge numerical relativity simulations. Our numerical modeling has allowed us to identify characteristic frequencies in the gravitational wave signal from the merged object, i.e., the short-lived hypermassive neutron star.

The expansion of the Universe redshifts the gravitational-wave frequencies, and the observed frequencies are lower in proportion to how fast the sources are moving away from us. Until recently, it was thought that gravitational-wave observations alone cannot determine this redshift since a degeneracy exists between the merger’s mass and its redshift: Only the redshifted mass—$M(1+z)$—is measured. However, measurements of the characteristic frequencies before and after the merger, together with prior knowledge of their true values from numerical simulations, allow us to break the mass-redshift degeneracy and extract the redshift directly from gravitational-wave observations. Measurements of the redshift alone enable crucial investigations of how the luminosity distance relates to redshift; this relationship in turn allows us to probe cosmological models. For relatively local sources, $z=0.01-0.04$, we recover redshift uncertainties of 10%–20%.

We have shown for the first time that there is a cosmological application for the postmerger signal and that redshift measurements can be made from binary neutron star merger signals. Our methodology adds to the handful of existing techniques that are currently known for using gravitational-wave signals as probes for cosmological inference.

Figure 1

Illustration of how the mass-redshift degeneracy is broken through the use of information from the inspiral and HMNS stage of a BNS merger event. Information on the redshifted mass as a function of the redshift (blue stripe) can be correlated with complementary information from the spectral properties of the HMNS phase. The overlap will provide a localized range in mass and redshift, breaking the degeneracy.

Figure 2

The normalized power-spectrum reference templates [Eq. (1)] for each of the five system masses as a function of frequency (black lines). Also plotted are the best-fit model templates defined in Eq. (2) (red dashed lines).

Figure 3

Measured best-fit values of $f_1$ and $f_2$ versus the total system mass (solid black circles). The vertical gray bars on the lower-frequency spectral feature represent the standard deviation of the Gaussian fit at each mass value (not the measurement uncertainty). The corresponding bars on the higher-frequency data represent the total frequency span of the feature. The blue curves and the red curves show the least-squares fit to $f_1$ and $f_2$ for a second- and first-order polynomial, respectively [given in Eq. (6)].

Figure 4

The joint posterior distributions on the redshift and total gravitational mass of the BNS system for single representative realizations of noise and system parameters. Each row of plots represents a simulated signal of one of the five system masses (see Table 1) ranging from low (bottom row) to high (top row) mass. Columns represent different simulated redshifts ranging from 0.01 (left) to 0.04 (right) in steps of 0.01. The green, blue, and red contours represent the posterior contributions from the inspiral measurement, the first HMNS spectral feature, and the second HMNS spectral feature, respectively. The black contours represent the final posterior distribution combining all measurements, and the black dots indicate the true simulated redshift and total mass values. In all cases, the contours enclose 68% of the probability. Overlapping regions have been filled according to the additive color system with the exception that regions outside all contours and the full interior of the final posterior contour have been left blank for clarity.